= Solution
Treat the two film surfaces as an effective interfacial tension $\Gamma=2\gamma$. The relevant <Bond number> for the gas-density difference is $\mathrm{Bo}=\Delta\rho gr^2/\Gamma$. Substituting the <Young–Laplace equation> and the preceding density relation gives
$$
\mathrm{Bo}=\frac{(4\rho_0\gamma/(rp_0))gr^2}{2\gamma}=\frac{2r}{h_0},\qquad h_0=\frac{p_0}{\rho_0g}.
$$
Thus the formal crossover $\mathrm{Bo}>1$ gives \b[$r>h_0/2\simeq4.25\,\mathrm{km}$]. This compares gravity on the excess <gas> with capillary restoration and neglects film weight. It does not say gravity-driven drainage of a real water film is negligible below kilometre scales. At this enormous radius, treating ambient pressure and density as uniform is itself no longer quantitatively reliable.
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